Order-reducing Form Symmetries and Semiconjugate Factorizations of Difference Equations · arXivDesk
0804.3579Apr 22, 200825 pages, 2 figures; reduction of order based on the new concept of form symmetry and semiconjugate factorization; Version 2 adds a converse to Theorem 2, a new example and several remarks; it also updates references (two new papers on this topic accepted) and corrects minor errors
Order-reducing Form Symmetries and Semiconjugate Factorizations of Difference Equations
The scalar difference equation xn+1=fn(xn,xn−1,...,xn−k)
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may exhibit symmetries in its form that allow for reduction of order through substitution or a change of variables. Such form symmetries can be defined generally using the semiconjugate relation on a group which yields a reduction of order through the semiconjugate factorization of the difference equation of order
k+1
into equations of lesser orders. Different classes of equations are considered including separable equations and homogeneous equations of degree 1. Applications include giving a complete factorization of the linear non-homogeneous difference equation of order
k+1
into a system of
k+1
first order linear non-homogeneous equations in which the coefficients are the eigenvalues of the higher order equation. Form symmetries are also used to explain the complicated multistable behavior of a separable, second order exponential equation.